Minimum-Phase/Allpass Decomposition

Every causal stable filter $H(z)$ with no zeros on the unit circle can be factored into a minimum-phase filter in cascade with a causal stable allpass filter:

$$H(z) \eqsp H_{\hbox{mp}}(z)\,S(z) \qquad\hbox{(Minimum-Phase/Allpass Decomposition)}$$

where $H_{\hbox{mp}}(z)$ is minimum phase, $S(z)$ is a stable allpass filter:

$$S(z) \eqsp \frac{s_L + s_{L-1}z^{-1}+ \cdots + s_1 z^{-(L-1)} + z^{-L}} {1 + s_1z^{-1}+ s_2 z^{-2}+ \cdots + s_L z^{-L}},$$

and $L$ is the number of maximum-phase zeros of $H(z)$ .

This result is easy to show by induction. Consider a single maximum-phase zero $\xi$ of $H(z)$ . Then $\left\vert\xi\right\vert>1$ , and $H(z)$ can be written with the maximum-phase zero factored out as

$$H(z) \eqsp H_1(z) (1-\xi z^{-1}).$$

Now multiply by $1=(1-\xi^{-1}z^{-1})/(1-\xi^{-1}z^{-1})$ to get

$$H(z) \eqsp \underbrace{H_1(z) (1-\xi^{-1}z^{-1})}_{\displaystyle\isdef H_2(z)} \underbrace{\frac{1-\xi z^{-1}}{1-\xi^{-1}z^{-1}}}_{\displaystyle\isdef S_1(z)}.$$

We have thus factored $H(z)$ into the product of $H_2(z)$ , in which the maximum-phase zero has been reflected inside the unit circle to become minimum-phase (from $z=\xi$ to $z=1/\xi$ ), times a stable allpass filter $S_1(z)$ consisting of the original maximum-phase zero $\xi$ and a new pole at $z=1/\xi$ (which cancels the reflected zero at $z=1/\xi$ given to $H_2(z)$ ).^12.2 This procedure can now be repeated for each maximum-phase zero in $H(z)$ .

In summary, we may factor maximum-phase zeros out of the transfer function and replace them with their minimum-phase counterparts without altering the amplitude response. This modification is equivalent to placing a stable allpass filter in series with the original filter, where the allpass filter cancels the maximum-phase zero and introduces the minimum-phase zero.

A procedure for computing the minimum phase for a given spectral magnitude is discussed in §11.7 below. More theory pertaining to minimum phase sequences may be found in [60].